Symmetry in Particle Physics From Circles to the Standard Model The Standard Model of particle physics is one of humanity’s greatest achievements. It describes three of the four fundamental forces and, as a result, it is our most complete understanding of nature. It gives us a framework within which all fun- damental interactions can be understood. Formulated in the 1960s and 1970s, it has proven itself time and again, with one exception: the Higgs boson, a crucial component of the theory, was nowhere to be found. Then, in July 2012, the European particle physics lab cern announced the discovery of the Higgs boson almost 50 years after its prediction, adding the nal missing piece of the universal puzzle. But what is the Higgs boson? To understand that, and the importance of its discovery, we need to understand the Standard Model itself. Fields and Forces To begin, there are four fundamental forces: the electromagnetic force, the strong force, the weak force and gravity. The Standard Model describes the rst three of these forces. The electromagnetic force is responsible for the familiar eects of electricity and magnetism, the strong force is responsible for the composition of subatomic particles (which are made of quarks), such as the proton and neutron inside the nucleus of atoms, and the weak force aects radioactive decay. The Standard Model uses quantum eld theory, a mathematical framework developed in the 20th century, to describe these forces in terms of elds. This idea is familiar to anyone who has played with iron lings and a magnet, where the lings fall into the shape of the eld lines. In physics, a eld is something that takes a value at every point in space and time. For example, consider the height of the surface of the sea. This takes a value (measured in metres, for instance) at every point on the sea, at any time. This is slightly dierent to the magnetic eld, whose value doesn’t just include the strength of the eld (analogous to the height of the sea) but also what direction it is pointing in. Physicists call the former a scalar eld and the latter a vector eld, because vectors are mathematical objects with magnitude and direction while scalars have only magnitude (like numbers). The strong, weak and electromagnetic forces of the Standard Model are all vector elds. 1 But how do the iron lings know how strong the eld is where they are, or which direction to point in? The eld must transmit this information to them somehow, i.e. it must be mediated by something. This is analogous to waves in the sea changing a buoy’s height. The ripples in the sea are, in quantum eld theory, considered as excitations of the eld. These excitations which mediate the force are called bosons — quantum particles which move around and interact with other particles, informing them of the eld strength. This identication of waves with particles is one of the profound results of quantum mechanics, called wave-particle duality. For electromagnetism, the boson is called the photon — the particle of light. The reverse must also be true: since excitations of elds can be thought of as particles, particles themselves must also be thought of as excitations of elds. Thus we have e.g. the electron eld permeating all of space, whose excitations are the particles we call electrons. One striking feature of these elds is that fundamentally they cannot be mea- sured. Instead, their various properties, such as charge, energy, etc., can be. For example, the fundamental eld associated with electromagnetism is called the vector potential A. The value of this eld is not a measurable quantity; instead we measure the values of the electric and magnetic elds and infer from these measurements the value of A. Symmetry Physicists have a formal notion of what a symmetry is. A square is symmetric because you can rotate it by 90 degrees and it will return to the same shape. Mathematically, we would describe this symmetry by listing all the operations we can do to it such that it remains the same. The list we end up with contains four rotations (by 90, 180, 270 and 360 degrees), and four reections (about the horizontal and vertical axes and about the diagonal lines from top-left to bottom- right and bottom-left to top-right). This list of operations is called a symmetry group. Mathematicians would say that this group is the symmetries of the square, named the dihedral group D4. But what of a circle? Rotating this shape by any angle at all is a symmetry, as is reecting it about any axis. So, the list of all the elements of this symmetry group is innite. Mathematicians call this group U(1). One of the hallmarks of modern physics is the use of symmetry. A central idea is that an experiment performed in some location should have the same result as an identical experiment performed elsewhere. That is, the result should be invariant under translation across space. The same is true across time. Thus, we say that theories should be translation-invariant. The same holds for rotations and, crucially, boosts in any direction. This means that an experiment in a car moving at a constant velocity (no acceleration) will yield the same results as a stationary one. This is called a Galilean transformation and the collection of all of 2 these operations together is called the Galilean group. Since physics is unchanged under the operations of the Galilean group, we say that this group is a symmetry. However, Galileo didn’t know about special relativity. This group of symmetries is actually only approximate, and doesn’t hold for high velocities and energies. The actual symmetry group is a generalisation of this one, called the Poincaré group. At low energies, where the speeds involved are much less than the speed of light, it reduces to the more familiar Galilean symmetries. Because of this, we say that the Poincaré group is a symmetry of physics. If we come up with a theory of nature, it has to be invariant under the operations of this group. Since these symmetries deal with space and time, we call them spacetime symmetries. But there are other symmetries — internal symmetries — disconnected from these external spacetime symmetries. Let’s return again to electromagnetism and the vector potential. Since this potential is responsible for the values of the electric and magnetic eld, it can be inferred from these values. However, it turns out that there are many possible potentials that would give rise to the same electric and magnetic elds. In fact, from a given potential there are a set of operations we can do to it to give us new potentials which generate identical electric and magnetic elds. This is a symmetry of electromagnetism, distinct from the spacetime symmetries of the Poincaré group. Any eld theory with this symmetry will be identical to electromagnetism, which means the symmetry in some sense denes electromagnetism. As it happens, this symmetry is exactly the same as the circle! This means we can say that electromagnetism is invariant under U(1) symmetry. Gauge Theory The most remarkable feature of this symmetry is that it in some sense depends on the location in space and time. In order to take this dependency into account, but still write down a theory that is U(1)-invariant, we must allow the vector potential to interact with matter, such as electrons. This is called gauge theory. It is responsible for electric charge, which determines the strength of the force on a charged particle in an electromagnetic eld. Bosons which mediate the force of these gauge elds are called gauge bosons. All fundamental forces have gauge symmetry (including gravity). It is believed to be a universal symmetry of nature, just like the symmetries of the Poincaré group. The gauge bosons of the strong force are called gluons, which bind the quarks together to form protons and neutrons. The symmetry involved is called SU(3), a more complicated group than U(1). One feature of U(1) is that the operations in the group commute. This is intuitively obvious, since if you rotate a shape by some angle, then again by any other angle, the result would be the same as rotating in the opposite order. The strong force does not have this property, which makes the interactions between gluons and quarks (and between gluons and themselves) 3 much more complex. The weak force has three gauge bosons, W +, W − and Z 0. The W ± bosons have electric charge and the Z 0 boson does not. This means that the weak force and electromagnetism must be related somehow. In fact, they are really manifestations of some larger force, the electroweak force. This has symmetry group SU(2) × U(1). The U(1) of electromagnetism is a subgroup of this. This is why the Standard Model of particle physics is sometimes called SU(3) × SU(2) × U(1). The product of these symmetry groups completely describes the fundamental forces of nature. Unfortunately, the nature they describe seems to be quite dierent to the world we live in. One problem concerns the mass of the W ± and Z 0 bosons. According to the above theory, these particles should be massless (like the photon, or all gauge bosons). Instead they have mass. Another problem is the dissimilarity between the weak force and electromagnetism in practice, despite them being in actuality part of the same thing. The phenomenon which solves these problems is called electroweak symmetry breaking. At low energies, the weak force and electromagnetism are actually separate.